Problem 58 - Precalculus

Question

Evaluate the logarithms using the change-of-base formula. Round to four decimal places. $$\log \sqrt{2} 9$$

Step-by-Step Solution

Answer

The logarithm \( \log_{\sqrt{2}} 9 \) is approximately 6.3402.

Step 1: Express the Logarithm

The expression \( \log_{\sqrt{2}} 9 \) indicates a logarithm with base \( \sqrt{2} \) and argument \( 9 \).

Step 2: Apply Change of Base Formula

To evaluate \( \log_{\sqrt{2}} 9 \), use the change of base formula: \( \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \). We can use any convenient base \( c \), such as \( 10 \) or \( e \) (natural logarithms). We'll choose base 10 for convenience, leading to the expression:\[\log_{\sqrt{2}} 9 = \frac{\log_{10} 9}{\log_{10} \sqrt{2}}\]

Step 3: Calculate \( \log_{10} 9 \)

Use a calculator to find \( \log_{10} 9 \). This gives approximately \( \log_{10} 9 \approx 0.9542 \).

Step 4: Calculate \( \log_{10} \sqrt{2} \)

Since \( \sqrt{2} = 2^{1/2} \), we have \( \log_{10} \sqrt{2} = \log_{10} (2^{1/2}) = \frac{1}{2} \log_{10} 2 \). Calculate \( \log_{10} 2 \approx 0.3010 \), hence \( \log_{10} \sqrt{2} \approx \frac{0.3010}{2} = 0.1505 \).

Step 5: Evaluate the Expression

Now, substitute the calculated values into the expression for \( \log_{\sqrt{2}} 9 \):\[\log_{\sqrt{2}} 9 = \frac{0.9542}{0.1505} \approx 6.3402\]

Step 6: Round the Result

The result is \( 6.3402 \) and it's already rounded to four decimal places as required.

Key Concepts

Understanding Logarithms

The concept of logarithms can be a bit tricky at first, but it's essentially about understanding the relationship between numbers in exponents. When you see a logarithm expressed as \( \log_a b \), it means you are trying to determine the power to which the base \( a \) must be raised to obtain \( b \). This function is particularly useful in mathematics because it helps reverse the process of exponentiation. To put it simply:

• The logarithm \( \log_a b \) equals \( c \) if and only if \( a^c = b \).
• It's an inverse operation to exponents.

Logarithms allow us to solve exponential equations more easily by transforming them into multiplication and division problems. This transformation is crucial for fields like computer science and engineering, where understanding growth rates or scaling is essential.

Delving into Exponents

Exponents play a fundamental role in the concept of logarithms. They provide a way to express repeated multiplication of a number, called the base. When you have a number like \( 2^3 \), it means you multiply 2 by itself three times: \( 2 \times 2 \times 2 = 8 \). Key points about exponents include:

• An exponent tells you how many times to use the base in a multiplication.
• Exponential growth or decay is used frequently in real-world applications, such as calculating interest rates or understanding population growth.

In the expression \( \log_{\sqrt{2}} 9 \), recognizing \( \sqrt{2} \) is equivalent to \( 2^{1/2} \) helps simplify calculations. As you break down these expressions further, understanding the exponents allows you to convert tricky expressions into manageable calculations.

Base 10 Logarithm - Common Logarithms

Base 10 logarithms, also known as common logarithms, are among the most widely used in everyday calculations. Often denoted simply as \( \log \) when the base is 10, these are essential for converting multiplication into addition, which can simplify many complex operations. When working with logarithms in base 10, calculators and computer systems are ready to handle these by default due to their versatility. Advantages of using base 10 logarithms include:

• Convenience: Most calculators have a \( \log \) function readily available.
• Simplicity: They streamline the process of dealing with large numbers or scales, such as in scientific notation.

In problems like evaluating \( \log_{\sqrt{2}} 9 \), using base 10 through the change of base formula facilitates easier calculation, especially when using technology tools. This ease of use makes base 10 logarithms indispensable in science and engineering.